Make The Camera Focus On A Rotating Cube On A Path In A Loop Animation
Never animate any object without a controler. 

In your case I would give a hierarchy that goes this way : A locator called MainCtrl => A locator called RotCtrl=> The Cube. 
MainCtrl would be the one constrained to the path and RotCtrl the one that would rotate the cube. The Cam could be the child of the MainCtrl. 
Then the camera could also have a Target (made with another locator) so that you could control how it looks to your cube during animation...

Whatever, NEVER animate directly your mesh, always use a control (rig) so you'll be able to change any geometry afterward without loosing animation